Introduction

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Introduction

• It’s a mathematical tool for signal decomposition, like Fourier’s Transform.
• Just as the Fourier transform decomposes a signal into a series of sine and cosine functions, the wavelet transform does so using a set of functions known as wavelets.
• Wavelets are functions generated by scaling and shifting a base function known as the mother wavelet.

Introduction

Introduction

• Morlet: Popular for time-frequency analysis in EEG and ECG.
• Mexican Hat (Ricker): Often used in spike detection in neural signals.
• Haar: Useful in quick decomposition of signals and feature extraction.
• Daubechies: Frequently used in ECG signal denoising and compression.
• Symlet: Another option for signal processing and feature extraction in EEG.
• Coiflet: Useful for denoising and baseline correction in biomedical signals.

Introduction

A wavelet transformation

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A wavelet transformation

Mathematical Expressions

The continuous wavelet transform of a signal \(f(t)\) is defined as:

\[W_f(a, b) = \int_{-\infty}^{\infty} f(t) \, \psi^*\left(\frac{t - b}{a}\right) \, dt\]

where:

• \(f(t)\) is the input signal,
• \(\psi\) is the mother wavelet,
• \(a\) is the scale parameter (controls the width of the wavelet),
• \(b\) is the translation parameter (controls the position of the wavelet),
• \(\psi^*\) denotes the complex conjugate of the mother wavelet. ## Mathematical Expressions

The continuous wavelet transform of a signal \(f(t)\) is defined as:

\[W_f(a, b) = \int_{-\infty}^{\infty} f(t) \, \psi^*\left(\frac{t - b}{a}\right) \, dt\]

where:

• \(f(t)\) is the input signal,
• \(\psi\) is the mother wavelet,
• \(a\) is the scale parameter (controls the width of the wavelet),
• \(b\) is the translation parameter (controls the position of the wavelet),
• \(\psi^*\) denotes the complex conjugate of the mother wavelet.

Mathematical Expressions

The inverse continuous wavelet transform is given by:

\[f(t) = \frac{1}{C_{\psi}} \int_{0}^{\infty} \int_{-\infty}^{\infty} W_f(a, b) \, \psi\left(\frac{t - b}{a}\right) \frac{db \, da}{a^2}\]

where:

where \(C_{\psi}\) is a normalization constant, defined as:

\[C_{\psi} = \int_{0}^{\infty} \frac{|\hat{\psi}(\omega)|^2}{\omega} \, d\omega\]

and \(\hat{\psi}(\omega)\) is the Fourier transform of the wavelet \(\psi(t)\).

Mathematical Expressions

The discrete wavelet transform decomposes the signal at discrete levels of scale. For a signal \(x[n]\), the wavelet decomposition is defined as:

\[c_{j, k} = \sum_{n} x[n] \, \psi_{j, k}(n)\]

where:

• \(\psi_{j, k}(n)= \frac{1}{\sqrt{2}}\psi\left(\frac{n-2^{j}k}{2^{j}}\right)\) represents the scaled and translated versions of the mother wavelet \(\psi\)
• \(j\) is the scale index, and \(k\) is the translation index.

The decomposition typically consists of approximation and detail coefficients at each scale:

Approximation coefficients \(a_j\): capture the low-frequency components. Detail coefficients \(d_j\) capture the high-frequency components.

Mathematical Expressions

The inverse discrete wavelet transform reconstructs the original signal from its approximation and detail coefficients: \[x[n] = \sum_{j} \sum_{k} c_{j, k} \, \psi_{j, k}(n)\]

This reconstruction process involves upsampling and filtering of the approximation and detail coefficients at each scale.

Using CWT

• Purpose: The CWT is used when you need a highly detailed, continuous analysis of a signal over all possible scales and positions.
• Output: CWT gives you a “heatmap” of wavelet coefficients, showing which frequencies (or scales) are present in the signal at each point in time. This allows for a continuous representation.
• Applications: CWT is useful for analyzing signals where you want to see the evolution of frequencies over time, such as:
  • Detecting subtle changes in frequencies over time (like brainwave analysis in EEG).
  • Signals with non-repeating, transient features (like spikes in biomedical signals, e.g., ECG).
• Trade-Off: CWT is more computationally expensive because it analyzes all scales and translations continuously. It gives lots of data but is slower and requires more memory.

Using DWT

• Purpose: The DWT is used when you want a compact, efficient representation of a signal, usually for compression or feature extraction. It analyzes only specific scales (powers of two), not continuously.
• Output: DWT gives you a set of coefficients at each level (or scale), capturing information at that specific scale. It’s efficient and uses fewer data points.
• Applications: DWT is ideal when you want to reduce the size of data or focus on a smaller set of frequencies, such as:
  • Image and audio compression (like JPEG 2000 or MP3 formats).
  • Feature extraction for machine learning (e.g., identifying specific patterns).
  • De-noising signals by discarding certain scales that contain noise.
• Trade-Off: DWT is computationally cheaper but less detailed than CWT. It doesn’t give a continuous heatmap but rather a discrete set of scales.

Introduction: Analyzing Complex Signals

• Signals like ECG are often non-stationary.
• Their frequency content and characteristics change over time.
• Traditional methods (e.g., Fourier Transform) analyze the signal globally, losing temporal information.
• Need a method to analyze signals at different time scales and frequency bands simultaneously.

What is Multiresolution Analysis (MRA)?

• A mathematical framework to decompose a signal into components at different levels of resolution or detail.
• Think of it like looking at a picture:
  • Coarse view (low resolution): See the overall structure.
  • Fine view (high resolution): See small details.
• MRA decomposes a signal into a series of approximations and details, capturing information across various scales.
• Often implemented using Wavelet Transforms (specifically the Discrete Wavelet Transform - DWT).

MRA: The Nested Subspace Concept

• MRA is built upon a sequence of nested subspaces \(V_j\) in \(L^2(\mathbb{R})\).
• These subspaces represent approximations of the signal at different resolutions.
• Key Properties:
  • Nesting: \(\dots \subset V_2 \subset V_1 \subset V_0 \subset V_{-1} \subset V_{-2} \subset \dots\)
  • Density: The union of \(V_j\) is dense in \(L^2(\mathbb{R})\).
  • Separation: The intersection of \(V_j\) is \(\{0\}\).
  • Scaling: \(f(t) \in V_j \iff f(2t) \in V_{j-1}\)

The Scaling Function (\(\phi\))

• Each subspace \(V_j\) is generated by scaled and translated versions of a single function called the scaling function, \(\phi(t)\).
• \(\{\phi(t-k) : k \in \mathbb{Z}\}\) forms an orthonormal basis for \(V_0\).
• \(\{\sqrt{2^j}\phi(2^j t-k) : k \in \mathbb{Z}\}\) forms an orthonormal basis for \(V_j\).
• The scaling function captures the “smooth” or low-frequency components – the approximation of the signal at a given scale.

The Wavelet Function (\(\psi\))

• The difference in information between two successive approximation spaces (\(V_j\) and \(V_{j-1}\)) is captured by the detail spaces, \(W_j\).
• \(V_{j-1} = V_j \oplus W_j\), where \(W_j\) is the orthogonal complement of \(V_j\) in \(V_{j-1}\).
• There exists a function \(\psi(t) \in W_0\), the wavelet function (or mother wavelet).
• \(\{\psi(t-k) : k \in \mathbb{Z}\}\) forms an orthonormal basis for \(W_0\).
• \(\{\sqrt{2^j}\psi(2^j t-k) : k \in \mathbb{Z}\}\) forms an orthonormal basis for \(W_j\).
• The wavelet function captures the “details” or high-frequency components.

Linking \(\phi\) and \(\psi\): Two-Scale Equations

• The scaling and wavelet functions are related through coefficients \(h_k\) and \(g_k\).
• \(\phi(t) = \sqrt{2} \sum_k h_k \phi(2t - k)\)
• \(\psi(t) = \sqrt{2} \sum_k g_k \phi(2t - k)\)
• These equations show how functions at one scale (left side) are constructed from functions at a finer scale (right side).
• The coefficients \(h_k\) and \(g_k\) are critical – they define the specific wavelet and are the impulse responses of filters used in discrete implementations.

From Math to Practice: Filters

• The sequences \(h_k\) and \(g_k\) correspond to digital filters:
  • \(H = \{h_k\}\) is a low-pass filter.
  • \(G = \{g_k\}\) is a high-pass filter.
• These filters are designed such that \(G\) is derived from \(H\) (for orthonormal wavelets, \(g_k = (-1)^k h_{N-1-k}\)).
• Applying the low-pass filter and downsampling corresponds to projecting the signal onto \(V_j\).
• Applying the high-pass filter and downsampling corresponds to projecting the signal onto \(W_j\).

Discrete MRA: Mallat’s Algorithm

• The discrete implementation of MRA is efficiently computed using Mallat’s Algorithm (or the Fast Wavelet Transform - FWT).
• It’s a pyramidal algorithm based on filter banks and downsampling/upsampling.
• Decomposes the discrete signal into approximation (\(A\)) and detail (\(D\)) coefficients at successive levels.

MRA Architecture

Why MRA for ECG?

• ECG signals are non-stationary and have features at different scales:
  • QRS complex: Sharp, high frequency, short duration.
  • P and T waves: Broader, lower frequency, longer duration.
  • Baseline wander: Very low frequency.
  • Muscle noise: High frequency.
• MRA naturally separates these components into different frequency bands (detail levels), making analysis easier.

Applications of MRA in ECG: Denoising

• Noise (powerline, muscle, baseline wander) occupies different frequency bands.
• Decompose noisy ECG using MRA.
• Noise components are isolated in specific detail coefficients:
  • High-frequency noise in fine detail levels.
  • Baseline wander in coarse approximation/detail levels.
• Apply thresholding to the noise-dominated coefficients (e.g., set small values to zero).
• Reconstruct the signal from the modified coefficients to get a denoised ECG.

Applications of MRA in ECG: Feature Extraction

• Different ECG waves are highlighted in different MRA levels:
  • QRS complex: Often prominent in mid-to-high frequency detail coefficients.
  • P and T waves: Appear in lower frequency detail coefficients or approximation coefficients.
• Analyze coefficients at relevant scales:
  • Detect QRS peaks by finding local maxima in specific detail levels.
  • Delineate P and T waves by analyzing the shape and zero-crossings of coefficients at coarser scales.
• Extract clinically relevant features (intervals, amplitudes) from the detected waves.

Other Applications in ECG

• Compression: MRA provides a sparse representation; many coefficients are small and can be discarded or quantized, reducing data size.
• Abnormality Detection and Classification: Features derived from the distribution or patterns of coefficients across different MRA levels can be used as input for machine learning algorithms to classify arrhythmias or other heart conditions.

Implementation: Wavelet Choice & Levels

• Wavelet Family: Daubechies (db) and Symlets (sym) are common for ECG. ‘db4’ is often cited for its similarity to the QRS complex.
• Number of Levels (\(J\)):
  • Higher levels give coarser approximations and lower frequency details.
  • Choose \(J\) so that key ECG features fall into specific, separable detail levels.
  • E.g., for a 360 Hz ECG, QRS (10-50 Hz) might be in D3-D5, P/T (1-10 Hz) in D5-D7, baseline wander (<1 Hz) in A7 or D8+.

Conclusion

• Multiresolution Analysis is a powerful technique for analyzing non-stationary signals like ECG.
• It decomposes the signal into different frequency bands/scales using scaling and wavelet functions.
• Implemented efficiently via Mallat’s algorithm (DWT).
• Enables effective ECG denoising by isolating noise at specific levels.
• Facilitates robust feature extraction (P, QRS, T) by highlighting them in different detail coefficients.
• A fundamental tool in modern automated ECG analysis systems.

ECG Analysis

# Elegir wavelet y nivel de descomposición
wavelet = "db4"
max_level = 3

# Descomposición multiresolución
coeffs = pywt.wavedec(ecg001, wavelet, level=max_level)
# coeffs[0]: coeficientes de aproximación al nivel 3 (cA3)
# coeffs[1:], coeficientes de detalle en niveles 3, 2, 1 (cD3, cD2, cD1)

# Reconstruir aproximación al nivel máximo
arr_approx = [coeffs[0]] + [np.zeros_like(c) for c in coeffs[1:]]
approx = pywt.waverec(arr_approx, wavelet)

ECG Analysis

# Reconstruir detalles por cada nivel
details = []
for i in range(1, max_level + 1):
    arr_detail = [np.zeros_like(c) for c in coeffs]
    arr_detail[i] = coeffs[i]
    detail = pywt.waverec(arr_detail, wavelet)
    details.append(detail)

ECG Analysis

ECG Analysis

ECG Analysis